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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

1 November 2026

2 essays on contacts that only push and how many answers

A cam cleat whose pin swings on a sprung link. A cam cleat at a lean of 2°, below its pin's lock lean of 2.56°, its pin carried on a link that swings about a frictionless pivot 2.2 pin heights above the rope, held by a spring. The link's lever is chosen so that, with the spring a quarter as stiff as the rope at that lever, cam and link turning together squeeze the rope 12.53 times the pull, the pin's own bottom edge. Along a path the pull rises to one, falls to nought and rises again. At this step the pull is 0: a fixed pin squeezes 12.53, this link 0.00, and a link at half that ratio 6.27.

Contacts that only push

A cleat on a swinging pin has two leans

Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.

5 figures

A growing four-bar walks away from its own double points. A four-bar with a rod of 3, its crank and rocker both of length t, pivoted so that both pivot circles pass through the slides' crossing, with the coupler point at (0.3, 0.2) in the rod's units. Drawn in units of t: the two pivot circles (dotted), their second meeting (ringed), the real coupler curve the machine draws (solid), and its real double points, filled where two real configurations meet and hollow where none reaches. At size t = 7 the curve's real double points are isolated (2.02), isolated (3.32), crossing (0.93); the number in brackets is how far the line of rod orientations through each passes from the unit circle's centre, a crossing below one and isolated above. As t grows the curve shrinks onto the crossing and the second meeting and the double points stay where they are.

How many answers

A growing four-bar reaches none of its departing double points

A four-bar growing into the elliptic trammel loses three double points, and wherever they are real the question is whether the machine passes through them. In the limit it passes through none: a configuration that reaches a point a crank's length away needs its rod along an isotropic direction, which no real rod has. How fast each point leaves the machine's reach is a ratio of four distances, √(1 + λ²)·|Z|·|Z − Zₛ| / (2b·|Z + 1|·|Z + iλ|), and it vanishes only at the pivot circles' second meeting. Every crossing is lost at a finite size: through a cusp, when the law predicts it, or by meeting another crossing and leaving as a complex pair, when it does not.

5 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

26 October 2026

16 essays on the problem backwards, members that pull, the shape is the unknown, one path to the tool, what a joint is, the chain before the lengths, motion that stops, drawn wrongly, the curve as an equation, more than one input, links with a width, machines you have met, teeth, the paths points trace, linkages and what can move

A constant pull holds the band for less — the problem backwards
A cam moves the law off the drums — members that pull
A relief is sized for one torque — the shape is the unknown
A seventh joint cannot move the elbow — one path to the tool
A tight cage spreads the load until the tracks join in — what a joint is
A V pressed onto its nose has a window — the chain before the lengths
A wheel held by its slots needs three pins — motion that stops
Six things a balancing spring is not — drawn wrongly
The quintic's cheap pivots lie off the line — the curve as an equation
The winning join is a gearbox too — more than one input
Two stations share a cut when they hand over together — links with a width
The first pull goes to the spring the lamp did without — machines you have met
The least line carries the root — teeth
Fit every extra slide at one pose — the paths points trace
Five degrees buy an even cut at a ratio of three — linkages
Put the motor where the rocking bar is longest — what can move

15 October 2026

14 essays on one path to the tool, what a joint is, the curve as an equation, how many answers, out of the plane, the motion, not the mechanism, machines you have met, many of one thing, wheels, and where they may not go, several legs, one platform, teeth, prescribed motion, linkages and what can move

A straight move ends at a stop — one path to the tool
A tight groove leans on one ball — what a joint is
The quintic's cheapness runs along a line — the curve as an equation
Three discs decide the real double points — how many answers
Three tuned joints stop at the fifth order — out of the plane
A crank that must come round picks another point — the motion, not the mechanism
A free length is absorbed where the springs attach — machines you have met
A grid that loses two crosses every cell — many of one thing
A load reads the road a radius cannot — wheels, and where they may not go
A seventh leg has no dead yaw of its own — several legs, one platform
The face carries the least line — teeth
A gentle return cannot widen a steep rise — prescribed motion
A ratio of three costs forty degrees — linkages
A drive that goes round is an angle — what can move

6 October 2026

16 essays on contacts that only push, the problem backwards, links with a width, the shape is the unknown, several legs, one platform, many of one thing, motion that stops, members that pull, the chain before the lengths, drawn wrongly, more than one input, machines you have met, the motion, not the mechanism, wheels, and where they may not go, prescribed motion and the paths points trace

A cleat set hard lets go easiest — contacts that only push
The arm belongs to the poses — the problem backwards
A cam machine is cut at its handovers — links with a width
A disc cut at its threshold grows a point — the shape is the unknown
A seventh leg must lean its own way — several legs, one platform
A spare is worth what its pretension leaves — many of one thing
A steered pin keeps only the window — motion that stops
A tolerance shortens the drum — members that pull
Return and stiffness come from the same aim — the chain before the lengths
Six things an indexer is not — drawn wrongly
The compound split has gearsets of its own — more than one input
The lamp needs three springs — machines you have met
The smoothest coupler point is chosen in time — the motion, not the mechanism
Three wheels read the road they measure — wheels, and where they may not go
A steep rise spends the arm's room — prescribed motion
A fourth block can only take freedom away — the paths points trace

3 October 2026

7 essays on what a joint is, how many answers, the curve as an equation, teeth, prescribed motion, linkages and what can move

A floating cage frees three balls and not six — what a joint is
A four-bar becomes a trammel along a family of laws — how many answers
Moving the pivot only adds terms — the curve as an equation
A helix pays for the root with face width — teeth
An arm that lifts against the cam can beat a slide — prescribed motion
A centred ram cannot be timed past its drag link — linkages
A push turns back at the crossing — what can move

1 October 2026

12 essays on the problem backwards, motion that stops, members that pull, the chain before the lengths, several legs, one platform, out of the plane, machines you have met, the motion, not the mechanism, more than one input, drawn wrongly, the shape is the unknown and many of one thing

A spring balanced against the load — the problem backwards
A curved slot moves the step — motion that stops
A drum for every law — members that pull
A seat with friction stays where it is pushed — the chain before the lengths
A seventh leg opens the dead yaw — several legs, one platform
A shake travels as a root of the play — out of the plane
A spring that must push is hung in the wrong place — machines you have met
No drive holds the third pole — the motion, not the mechanism
One cross-ratio decides the compound split — more than one input
Six things a cam drive is not — drawn wrongly
The flank that trims the root — the shape is the unknown
Twelve cables survive a slack one — many of one thing

29 September 2026

30 essays on the problem backwards, the chain before the lengths, contacts that only push, the curve as an equation, members that pull, the shape is the unknown, what a joint is, wheels, and where they may not go, motion that stops, the motion, not the mechanism, how many answers, links with a width, prescribed motion, teeth, linkages, the paths points trace, more than one input, drawn wrongly, many of one thing, out of the plane, several legs, one platform, machines you have met and what can move

The slide five poses almost have — the problem backwards
A contact turns and slides at once — the chain before the lengths
A hold the load squeezes shut — contacts that only push
The cheapest curves at each degree — the curve as an equation
A cone drive is not two cones — members that pull
A disc whose roots are not sockets — the shape is the unknown
A Rzeppa cage adds its clearances — what a joint is
A second wheel cannot correct the first — wheels, and where they may not go
A V holds the wheel to first order — motion that stops
The third pole goes anywhere but the circle — the motion, not the mechanism
The trammel takes both double points — how many answers
Three arcs, cut in the right places — links with a width
A cam on a chain steps at every tooth — prescribed motion
Four knobs for the root — teeth
A centred slide has a best rod, and an offset beats it — linkages
A second slide keeps only the rod — the paths points trace
A second planetary lets the variator carry a difference — more than one input
A stiff pin buys grip with release — contacts that only push
Six things a power split is not — drawn wrongly
A real road sets its own power — wheels, and where they may not go
Four cuts serve the whole cell — links with a width
A third slide moves only through the crossing — the paths points trace
The load picks the wall — the problem backwards
A cable is half a brace — many of one thing
A shake has an order — out of the plane
A stop aimed at its pole holds by curvature — the chain before the lengths
Four terms of eighteen draw the quintic — the curve as an equation
No path across the dead yaw misses it — several legs, one platform
Two springs stand in for a parallelogram — machines you have met
The price is one self-stress — what can move

27 September 2026

10 essays on contacts that only push, out of the plane, members that pull, several legs, one platform, machines you have met, the motion, not the mechanism, wheels, and where they may not go, prescribed motion, teeth and linkages

Two jaws and a coefficient — contacts that only push
A third leg that keeps the shake — out of the plane
One belt for every step — members that pull
The dead yaw opens into a wedge — several legs, one platform
The lamp that stays where it is put — machines you have met
The other pole is on the circle — the motion, not the mechanism
What a wheel counts on a rough road — wheels, and where they may not go
A cam behind a bent joint — prescribed motion
The shift that evens the wear — teeth
The first stage does not know what it drives — linkages

26 September 2026

5 essays on machines you have met, how many answers, what a joint is, the paths points trace and what can move

The horsehead is a drum — machines you have met
The double point a slide cannot keep — how many answers
Two errors, one ellipse — what a joint is
A slide exchanges the crank and the rod — the paths points trace
The price of the other family — what can move

24 September 2026

7 essays on out of the plane, what a joint is, the shape is the unknown, motion that stops, many of one thing, more than one input and links with a width

A shared load is a shared motion — out of the plane
Constant velocity is a mirror — what a joint is
A clearance leaves one contact — the shape is the unknown
A flat holds the wheel to second order — motion that stops
A seam with no bar missing — many of one thing
Every power split is on one curve — more than one input
Four arcs and the false alarm is gone — links with a width

23 September 2026

7 essays on the problem backwards, many of one thing, how many answers, prescribed motion, the paths points trace, linkages and what can move

The slide a fourth pose leaves — the problem backwards
An edge that knows which side it is on — many of one thing
The double points that leave for infinity — how many answers
Where on the sine the arm swings — prescribed motion
A closed form belongs to a rotation — the paths points trace
Fifteen pairs and one cut — linkages
The clearance that makes the framework generic — what can move

20 September 2026

9 essays on the shape is the unknown, the curve as an equation, motion that stops, more than one input, several legs, one platform, members that pull and as built

A twist steadies what it cannot tighten — the shape is the unknown
The price is on the equation — the curve as an equation
Two circles for the price of one — the curve as an equation
What a second contact is for — the shape is the unknown
The disc decides the pin count — motion that stops
The power that goes round twice — more than one input
Tilted, near the dead yaw — several legs, one platform
Which walls a strand is held by — members that pull
A piano hinge is not forty door hinges — as built

17 September 2026

17 essays on the problem backwards, what a joint is, members that pull, more than one input, many of one thing, several legs, one platform, contacts that only push, out of the plane, motion that stops, how many answers, wheels, and where they may not go, the chain before the lengths, the motion, not the mechanism, prescribed motion, linkages, the paths points trace and what can move

Where a pin becomes a slide — the problem backwards
A coupling that only translates — what a joint is
A strand in a tube — members that pull
Sliding the travel across the pole — more than one input
The count says how many and not where — many of one thing
The dead yaw is a design choice — several legs, one platform
The hold is in the corners — contacts that only push
The lines a leg turns about and the lines it is pushed along — out of the plane
When the index law becomes a choice — motion that stops
Where a slide puts the rest of the degree — how many answers
A bearing is a planetary with no teeth — wheels, and where they may not go
A slide turns nothing — the chain before the lengths
The frame seen from the coupler — the motion, not the mechanism
The time a crossover takes — prescribed motion
A quick return that cuts evenly — linkages
The dip that buys the dwell — the paths points trace
Every change point lies flat — what can move

15 September 2026

12 essays on teeth, links with a width, the motion, not the mechanism, members that pull, out of the plane and many of one thing

The angle the standard left free — teeth
The gap is a straight line in the metal — links with a width
The linkage, put back from two curves — the motion, not the mechanism
The wrap that walks along the axis — members that pull
What a leg of three joints leaves free — out of the plane
Which diagonal rigidifies a grid — many of one thing
A tooth that lives on a sphere — teeth
The regions overlap and the parts never meet — links with a width
The two numbers are the curves' own — the motion, not the mechanism
A link may be bent — links with a width
The mesh with one curvature reversed — teeth
Contact that runs along the tooth — teeth

14 September 2026

11 essays on motion that stops, more than one input, prescribed motion, several legs, one platform, the shape is the unknown, the paths points trace, how many answers, linkages and what can move

Two pins and no dwell at all — motion that stops
A bounded ratio made unbounded — more than one input
A flat face on an arm is worse — prescribed motion
A ratchet with no teeth — motion that stops
One placement of every placement — several legs, one platform
The clearance that is the seal — the shape is the unknown
The kind is decided before the lengths are — the paths points trace
The mesh inside keeps the half — how many answers
Three rotations, and four benches — linkages
A length error is undone by its own size — what can move
The right angle as a tolerance — what can move

12 September 2026

13 essays on the paths points trace, prescribed motion, linkages, how many answers, what can move, the shape is the unknown and several legs, one platform

Where three machines keep one area — the paths points trace
An arm is an offset that grows with the lift — prescribed motion
Four kinds of slider-crank — linkages
Nine times through each circular point — how many answers
The flattest dwell is not the longest — the paths points trace
A drag link ahead of a crank-rocker — linkages
A null space of fifteen is not noise — the paths points trace
A parallelogram a micron wrong — what can move
A roller in a groove changes walls with the speed — prescribed motion
Every rational gear ratio has a degree — how many answers
Rotors that mesh and cannot drive each other — the shape is the unknown
Two orientations no position can rescue — several legs, one platform
Nine bars that ought to be rigid — what can move

11 September 2026

18 essays on the paths points trace, prescribed motion, how many answers, several legs, one platform, linkages and what can move

The curve the other assembly draws — the paths points trace
A point the machine never reaches — the paths points trace
A sextic that comes apart — the paths points trace
A symmetric curve from a lopsided machine — the paths points trace
An offset trades the rise for the return — prescribed motion
Never three circuits — how many answers
Round a cusp into another assembly — several legs, one platform
A degree counted on a line — how many answers
A flat face asks for a convex cam — prescribed motion
Eight kinds of four-bar — linkages
The area a coupler point encloses — the paths points trace
The smallest parallel robot — several legs, one platform
A cam that holds its follower both ways — prescribed motion
A swing and a time ratio — linkages
The curve nobody eliminates — how many answers
What a fourth leg buys — several legs, one platform
A yaw that is singular everywhere — several legs, one platform
One freedom, and a motion that never repeats — what can move

5 September 2026

50 essays on numbers that were measured, prescribed motion, several legs, one platform, linkages, out of the plane, teeth, as built, the motion, not the mechanism, more than one input, contacts that only push, wheels, and where they may not go, the problem backwards, links with a width, members that pull, the curve as an equation, the chain before the lengths, one path to the tool and drawn wrongly

A dimension is a measurement — numbers that were measured
The matrix a calibration inverts — numbers that were measured
The direction no protractor can see — numbers that were measured
The coordinates the site already had — numbers that were measured
A ruler and a protractor — numbers that were measured
How many poses are enough — numbers that were measured
Where a calibration should measure — numbers that were measured
A lift is a size and a law is a shape — prescribed motion
A platform that measures itself — several legs, one platform
What another measurement is worth — numbers that were measured
Four indices, four answers — numbers that were measured
Grashof is a shape test — linkages
Bennett's condition is a ratio — out of the plane
The module is a size, the ratio is a shape — teeth
The pose the machine cannot reach — numbers that were measured
A band with a direction in it — as built
A curvature is a size with a minus sign — the motion, not the mechanism
Every length wrong, every reading right — numbers that were measured
The transmission angle has no size — more than one input
A cone has no size — contacts that only push
A wheel that cannot report its radius — wheels, and where they may not go
The instrument's error, multiplied — numbers that were measured
What a synthesis assumes it knows — the problem backwards
Where the boundary moved again — as built
A body is all size — links with a width
A drum is a size, a wrap is a shape — members that pull
Reading a residual — numbers that were measured
A compiled machine and its own scale — the curve as an equation
A parameter the model has not got — numbers that were measured
A graph has no numbers at all — the chain before the lengths
One set of lengths, two machines — numbers that were measured
Three machines, one curve — numbers that were measured
An arm's parameters and its poses — one path to the tool
Nine parameters, two of them invisible — numbers that were measured
Six things a measurement cannot tell you — drawn wrongly
A machine that measures itself — numbers that were measured
Where a length comes from — numbers that were measured
Where the two analyses cross — numbers that were measured
The same part, dimensioned twice — numbers that were measured
Which feature to hold tight — numbers that were measured
The common normal, and where it is — numbers that were measured
A number that runs away — numbers that were measured
The chart breaks, the machine does not — numbers that were measured
Six per joint is two too many — numbers that were measured
Which numbers have a size — numbers that were measured
A count is neither — numbers that were measured
A calibration is a synthesis with more equations — numbers that were measured
Two instruments disagree about the worst — numbers that were measured
What a model is allowed to change — numbers that were measured
What this field cannot measure — numbers that were measured

1 September 2026

20 essays on links with a width, prescribed motion, the problem backwards, many of one thing, what a joint is, drawn wrongly, one path to the tool and as built

A link that takes up room — links with a width
A gap is a number — links with a width
A shape with a dent in it — links with a width
A gap with corners in it — links with a width
A sweep that missed nothing — links with a width
Two bars that have to cross — links with a width
A follower needs a face — prescribed motion
A plane is a colour — links with a width
The room a machine sweeps — links with a width
The hole the machine needs — links with a width
The crank that cannot turn all the way — links with a width
Free space comes in pieces — links with a width
A pin is not a point — links with a width
A defect that is not kinematic — the problem backwards
A stack that has to fit — many of one thing
The block in the guide has a length — what a joint is
Six things a body is not — drawn wrongly
The arm that hits itself — one path to the tool
Where the boundary moved — as built
A clearance inside a tolerance box — as built

31 August 2026

20 essays on the curve as an equation, the paths points trace, how many answers, what can move, the problem backwards, the chain before the lengths, drawn wrongly and as built

A demand that is an equation — the curve as an equation
Every curve is a sum of cosines — the curve as an equation
A circle costs one term — the curve as an equation
Four bars that add two angles — the curve as an equation
A parallelogram carries an angle, and only so far — the curve as an equation
Three linkages, one equation — the paths points trace
Doubling is cheaper than adding — the curve as an equation
The machine, compiled — the curve as an equation
Five bars for a line, four hundred for a quintic — the curve as an equation
The equation a four-bar satisfies — how many answers
The proof drew more than the curve — the curve as an equation
A bar between two midpoints — the curve as an equation
One freedom and four hundred links — what can move
Where the machine stops being the function — the curve as an equation
Exact costs more than close — the curve as an equation
Prescribing a curve rather than points — the problem backwards
What universality is worth — the curve as an equation
A machine with one dyad in it — the chain before the lengths
Six things a compiled linkage is not — drawn wrongly
Exactness a micron destroys — as built

30 August 2026

20 essays on what a joint is, several legs, one platform, what can move, out of the plane, wheels, and where they may not go, drawn wrongly, one path to the tool and as built

A joint is a surface that slides on itself — what a joint is
Six, and no others — what a joint is
The count cannot tell a pin from a slide — what a joint is
Twelve kinds of freedom — what a joint is
Almost nothing is a group — what a joint is
What a point sees — what a joint is
A chain multiplies — what a joint is
Three legs and one plane — several legs, one platform
Four joints that give a group, and four that do not — what a joint is
Legs intersect — what a joint is
The freedom that is a set — what can move
Two planes meeting in a line — what a joint is
Compose two positions and see where you land — what a joint is
A name for each overconstraint — out of the plane
The instrument that is not a derivative — what a joint is
A higher pair has no group — what a joint is
One bracket, two subjects — wheels, and where they may not go
Six things a joint is not — drawn wrongly
The arm that is a group — one path to the tool
The pair a catalogue sells — as built

28 August 2026

20 essays on the chain before the lengths, how many answers, what can move, the problem backwards, out of the plane, drawn wrongly, one path to the tool and as built